28 research outputs found
Game-theoretic approach to risk-sensitive benchmarked asset management
In this article we consider a game theoretic approach to the Risk-Sensitive
Benchmarked Asset Management problem (RSBAM) of Davis and Lleo \cite{DL}. In
particular, we consider a stochastic differential game between two players,
namely, the investor who has a power utility while the second player represents
the market which tries to minimize the expected payoff of the investor. The
market does this by modulating a stochastic benchmark that the investor needs
to outperform. We obtain an explicit expression for the optimal pair of
strategies as for both the players.Comment: Forthcoming in Risk and Decision Analysis. arXiv admin note: text
overlap with arXiv:0905.4740 by other author
Optimal co-adapted coupling for the symmetric random walk on the hypercube
Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class
Conditioning an additive functional of a markov chain to stay non-negative. I, Survival for a long time
Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E -> R \ {0}, and let (phi(t))(t >= 0) be an additive functional defined by phi(t) = integral(0)(t)(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) is nonnegative until time T and prove weak convergence of the conditioned process as T -> infinity
Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level
Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v: E -> R \ {0}, and let (phi(t))(t >= 0) be defined by phi(t) = integral(0)(t) v(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) hits level y before hitting 0 and prove weak convergence of the conditioned process as y -> infinity. In addition, we show the relationship between the conditioning of the process (phi(t))(t >= 0) with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (phi(t))(t >= 0) with a negative drift to drift to infinity and the conditioning of it to hit large levels before hitting 0
On the compensator in the Doob-Meyer decomposition of the Snell envelope
Let be a semimartingale, and its Snell envelope. Under the assumption that , we show that the finite-variation part of is absolutely continuous with respect to the decreasing part of the finite-variation part of . In the Markovian setting, this enables us to identify sufficient conditions for the value function of the optimal stopping problem to belong to the domain of the extended (martingale) generator of the underlying Markov process. We then show that the \textit{dual} of the optimal stopping problem is a stochastic control problem for a controlled Markov process, and the optimal control is characterised by a function belonging to the domain of the martingale generator. Finally, we give an application to the smooth pasting condition
On the policy improvement algorithm in continuous time
We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems for continuous-time processes. The main results assume only that the controls lie in a compact metric space and give general sufficient conditions for the PIA to be
well-defined and converge in continuous time (i.e. without time discretisation). It emerges that the natural context for the PIA in continuous time is weak stochastic control. We give examples
of control problems demonstrating the need for the weak formulation as well as diffusion-based classes of problems where the PIA in continuous time is applicable